Hybridizing ant colony optimization algorithm with nonlinear programming method for effective optimal design of sewer networks

Abstract

The ant colony optimization algorithm (ACOA) is hybridized with nonlinear programming (NLP) for the optimal design of sewer networks. The resulting problem is a highly constrained mixed integer nonlinear problem (MINLP) presenting a challenge even to the modern heuristic search methods. In the proposed hybrid method, The ACOA is used to determine pipe diameters while the NLP is used to determine the pipe slopes of the network by proposing two different formulations. In the first formulation, named ACOA-NLP1, a penalty method is used to satisfy the problem constraints while in the second one, named ACOA-NLP2, the velocity and flow depth constraints are expressed in terms of the slope constraints which are easily satisfied as box constraint of the NLP solver leading to a considerable reduction of the search space size. In addition, the assumption of minimum cover depth at the network inlets is used to calculate the nodal cover depths and the pump and drop heights at the network nodes, if required, leading to a complete solution. The total cost of the constructed solution is used as the objective function of the ACOA, guiding the ant toward minimum cost solutions. Proposed hybrid methods are used to solve three test examples, and the results are presented and compared with those produced by a conventional application of ACOA. The results indicate the effectiveness and efficiency of the proposed formulations and in particular the ACOA-NLP2 to optimally solve the sewer network design optimization problems. PRACTITIONER POINTS: ACOA is hybridized with NLP for the effective optimal design of sewer networks. Here, ACOA is used to determine pipe diameters and NLP is used to determine the network pipe slopes with predefined pipe diameters. In ACOA-NLP1, a penalty method is used to enforce the problem constraints. In ACOA-NLP2, velocity and flow depth constrains are expressed in terms of slope constraint.